Rainbow Gauge Theory (Honeycomb Loop Model)

Edges represent spins, with colored lines representing the nontrivial states. Each Monte Carlo step flips all six edges around a random hexagon, accepted with the Glauber probability 1/(1+exp(ΔE/T)). Because every vertex of the honeycomb has only three bonds, each color must appear zero or two times there, so loops of different colors can never cross.

A hexagon is flippable when at most one nontrivial color appears on its six edges — only those hexagons admit a legal move. Positive flippability f rewards configurations with many flippable hexagons, keeping the dynamics loose; negative f rewards jammed configurations where most hexagons are blocked by two colors meeting.